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Reconstructing Ancestral Genomic Sequences
by Co-Evolution: Formal Definitions, Computational
Issues, and Biological Examples
*TAMIR TULLER,1,2 *HADAS BIRIN,3 MARTIN KUPIEC,4 and EYTAN RUPPIN3,5
ABSTRACT
The inference of ancestral genomes is a fundamental problem in molecular evolution. Due tothe statistical nature of this problem, the most likely or the most parsimonious ancestralgenomes usually include considerable error rates. In general, these errors cannot be abol-ished by utilizing more exhaustive computational approaches, by using longer genomicsequences, or by analyzing more taxa. In recent studies, we showed that co-evolution is animportant force that can be used for significantly improving the inference of ancestralgenome content. In this work we formally define a computational problem for the inferenceof ancestral genome content by co-evolution. We show that this problem is NP-hard andhard to approximate and present both a Fixed Parameter Tractable (FPT) algorithm, andheuristic approximation algorithms for solving it. The running time of these algorithms onsimulated inputs with hundreds of protein families and hundreds of co-evolutionary rela-tions was fast (up to four minutes) and it achieved an approximation ratio of <1.3. We useour approach to study the ancestral genome content of the Fungi. To this end, we implementour approach on a dataset of 33, 931 protein families and 20, 317 co-evolutionary relations.Our algorithm added and removed hundreds of proteins from the ancestral genomes in-ferred by maximum likelihood (ML) or maximum parsimony (MP) while slightly affectingthe likelihood/parsimony score of the results. A biological analysis revealed various pieces ofevidence that support the biological plausibility of the new solutions. In addition, we showedthat our approach reconstructs missing values at the leaves of the Fungi evolutionary treebetter than ML or MP.
Key words: Co-evolution, maximum likelihood, maximum parsimony, reconstruction of ances-
tral genomes.
1Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, Israel.2Department of Molecular Genetics, Weizmann Institute of Science, Rehovot, Israel.3School of Computer Science, Tel Aviv University, Tel Aviv, Israel.4Department of Molecular Microbiology and Biotechnology, Tel Aviv University, Tel Aviv, Israel.5School of Medicine, Tel Aviv University, Tel Aviv, Israel.*T.T. and H.B. contributed equally to this work.
JOURNAL OF COMPUTATIONAL BIOLOGY
Volume 17, Number 9, 2010
# Mary Ann Liebert, Inc.
Pp. 1327–1344
DOI: 10.1089/cmb.2010.0112
1327
1. INTRODUCTION
The problem of reconstructing ancestral genomic sequences is as old as the field of molecular
evolution. The first approach for inferring ancestral genomic sequences was suggested by Fitch around
40 years ago (Fitch, 1971). This first algorithm assumed a binary alphabet, and was based on the Maximum
Parsimony (MP) criteria, i.e., find the labels to the internal nodes of a tree that minimize the number of
mutations along the tree edges. Over the years this basic algorithm was generalized in many ways. Sankoff
showed how to efficiently solve versions of the maximum parsimony problem with a non-binary alphabet and
with multiple edge weights (Sankoff, 1975). Similar algorithms for inferring ancestral sequences based on
maximum likelihood (ML; instead of maximum parsimony) were suggested more than 15 years later (Barry
and Hartigan, 1987; Pupko et al., 2000; Elias and Tuller, 2007; Felsenstein, 1993; Krishnan et al., 2004;
Pagel, 1999). Recently, similar approaches were used to infer ancestral sequences in phylogenetic networks
( Jin et al., 2006).
Dozens of biological studies have dealt with the reconstruction of ancestral genomic sequences and
ancestral genomes. For example, reconstruction of ancestral sequences was used for understanding the
origins of genes and proteins (Zhang and Rosenberg, 2002; Thornton et al., 2003; Tauberberger et al., 2005;
Jermann et al., 1995; Hillis et al., 1994; Gaucher et al.; 2003, Cai et al., 2004; Blanchette et al., 2004;
Ouzounis et al., 2006), and for aligning genomic sequences (Hudek and Brown, 2005), for inferring
ancestral enzymes and genomes (Yang et al., 1995; Koshi and Goldstein, 1996; Rascola et al., 2007; Ma
et al., 2006; Csuros and Miklos, 2009; Cohen et al., 2008), noncoding genomic regions (Gorbunov et al.,
2009), and SNPs (Hacia et al., 1999).
The main problem related to reconstructing ancestral sequences and genomes is that in practice many
times the reconstructed sequences contain a large number of errors. A major source of this phenomenon is
the existence of multiple local and/or global maxima in the solution space searched by both the maximum
likelihood and the maximum parsimony approaches (Chor et al., 2000; Tuller et al., 2009a). Thus, many
times our confidence in the most likely or most parsimonious reconstructed ancestral state is not too high.
As the above mentioned algorithms assume that different sites and different genes/proteins evolve inde-
pendently, this problem cannot be solved by adding more samples or more taxa (Li et al., 2008).
Several studies demonstrated that functionally and physically interacting proteins tend to co-evolve
(Juan et al., 2008; Sato et al., 2003, 2005; Tuller et al., 2009b; Felder and Tuller, 2008), and that
co-evolutionary relations between proteins are quite ubiquitous (Wapinski et al., 2007). Some of these
previous investigations used the fact that interacting proteins have correlative evolution in order to suc-
cessfully predict physical interactions ( Juan et al., 2008; Sato et al., 2003, 2005).
Based on these results, we recently suggested a different approach, the Ancestral Co-Evolver (ACE), for
improving the accuracy of reconstructed ancestral genomes (Tuller et al., 2009a). Our approach was based
on utilizing information embedded in the co-evolution of functionally/physically interacting proteins. We
used this approach to study the genome content of the Last Universal Common Ancestor (LUCA). In this
work we give a formal description of the ancestral co-evolution problem, we analyze its computational
complexity and describe algorithms for solving it; the performances of these algorithms are demonstrated
by a simulation study. Additionally, we use the ACE for studying a new biological example, the ancestral
genome content of the Fungi; specifically, we show that our approach reconstructs missing values at the
leaves of the Fungi evolutionary tree better than ML or MP.
2. DEFINITIONS AND PRELIMINARIES
For simplicity, we assume a binary alphabet. However, all the results here can be easily generalized to
models with more than two characters. In this work we assume that in general, if they do not have co-
evolutionary relation, neighbor sites in the input sequences evolve independently. Thus, the basic com-
ponents in the model and algorithms are single characters.
Our goal is to reconstruct the ancestral states for a set of organisms T of size jT j ¼ n. A phylogenetic
tree is a rooted binary tree T¼ (V(T), E(T)) together with a leaf labeling function l, where V(T) is the set of
vertices and E(T) the set of edges. In our context, a weight table is attributed to each edge (u, v)¼ e 2 E(T).
This weight table includes a weight (a positive real number) for each pair of labels of the two vertices (u,
v)¼ e.
1328 TULLER ET AL.
In this work, we assume that each node in a phylogenetic tree corresponds to a different organism. The
leaves in a phylogenetic tree correspond to organisms that exist today (T ), while the internal nodes
correspond to organisms that have become extinct (T 0). Thus, the leaf labeling function is a bijection
between the leaf set L(T) and the set of organisms that exist today, TIn our binary case, each label is a binary sequence; all the sequences have the same length. In the case of
conventional ML/MP, as we assume an i.i.d. case where different characters in a sequence evolve inde-
pendently, we can describe the algorithm for sequences of lengths one: i.e., either ‘‘1’’ or ‘‘0’’. A full
labeling of a phylogeny kk(T) is a labeling of all nodes of the tree such that the labels at the leaves are the
same as in l, i.e., for all l 2 L(T) k(l)¼ kk(l).
We can name each node after its corresponding organism. Let OT( � ) denote a function that returns the
index of the organisms corresponding to each node in T, i.e., for every v 2 V(T), OT(v) is the index of the
organism (from T [ T 0) corresponding to v.
A co-evolving forest F¼ (SF ¼fT1, T2, . . .g, Ec(SF)) is a set of phylogenetic trees, SF, with identical
topology that correspond to the same organisms [i.e., each tree has the same O( � )], and an additional set of
edges, EC(SF), that connect pairs of nodes in different trees. This set of edges represents the coevolutionary
relations between pairs of protein families. Edges in EC(SF) must connect pairs of nodes that correspond to
the same organism (i.e., (v, u) 2 Ec(SF), v 2 V(T1), u 2 V(T2)) OT1(v)¼OT2
(u); Fig. 1); we call such
pairs of nodes legal co-evolutionary pairs.
The edges in Ec (SF) are named co-evolution edges while edges that are part of the evolutionary trees are
named tree edges. For example, Figure 1A includes a co-evolving forest with two trees (the co-evolution
edges are dashed with arrows while the tree edges are continuous). As co-evolutionary relations. In this
work we assume that new co-evolutionary edges do not appear/dissapear during evolution. Namely, we
assume that if there is a co-evolutionary edge between a legal co-evolutionary pair of nodes in two trees
than all the legal co-evolutionary pairs of nodes in the two trees are connected by co-evolutionary edge.
A full labeling of a co-evolving forest kk(SF) is a full labeling, fkk(T1), kk(T2), . . .g, of all the nodes of the
trees in SF. The roots of a co-evolving forest are the set of roots of the phylogenetic trees in the co-evolving
forest.
As mentioned, a co-evolving forest also includes a weight table for each co- evolution edge and each tree
edge. These weight tables are cost functions that return a real positive number for each pairs of labels at the
two ends of the edge. In the case of tree edges, these weights reflect the probability of a mutation along the
1 0 0
0 0
1 0 1x1 y1
x2 y2
0 0 0 1 1 0 1 1
w 1
w 2
w 3
w 4
0 0
x1 x2
0 0 0 1 1 0 1 1
v 1
V2
V3
V4
x2
0
0 0
1
1
T1 T2
T3
T1
T2
T3
B
A x3
x4 y2
y3
y4
FIG. 1. (A) A simple example of a co-evolving forest with three trees, and one co-evolution edge connecting node x2
in tree T1 and node y2 in tree T2; the weight table corresponding to this co-evolution edge is in red. The weight table
corresponding to the tree edge (x1, x2) in T1 is in green. (B) The co-evolutionary graph corresponding to the co-evolving
forest in A.
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1329
edge. In the case of co-evolution edges, these weights reflect the distribution of mutual occurrence of the
labels of the nodes at the ends of the edge.
This leads us to the formal definition of the problem we are concerned with, the Ancestral co-evolution
problem:
Problem 1. Ancestral co-evolution (Ancest–co–evol)
Input: A co-evolving forest, F¼ (SF, EC (SF)), and a real number, B.
Question: Are there labels for the internal nodes of all the trees in the co-evolving forest such that the
sum of the corresponding weights along all the tree edges and the co-evolution edges is less than B?
The following example demonstrates the advantages of the ancestral co-evolver compared to the simple
i.i.d parsimony approach.
Example 1. Consider the co-evolving forest that appears in Figure 1, and assume that all the weight tables
of the tree edges are identical to the table that appears in the figure where [v1, v2, v3, v4]¼ [0, 1, 1, 0]. It is easy
to see that there are two MP solutions for the labels of the internal nodes in the phylogenetic tree T1: either [x1,
x2, x3, x4]¼ [0, 0, 0, 0] or [x1, x2, x3, x4]¼ [0, 1, 1, 1] gives the same score (2). In the case of the tree T2, it is easy
to see that there is one MP solution: all the labels of the internal nodes are ‘‘0.’’ Now, suppose that in the weight
table corresponding to the co-evolution edge (x2, y2), w1 is the smallest entry. Thus, by co-evolution study, the
solution [x1, x2, x3, x4]¼ [0, 0, 0, 0] is more plausible and the ambiguity in the labels of T1 is solved.
Note that in general it is not necessarily required that the solution of each tree separately will be the most
parsimonious (see the next sections). The minimal sum of edge weights corresponding to a co-evolving
forest, F (problem 1) is named the cost of F. A co-evolutionary graph is an undirected graph that describes
the co-evolution edges in the co-evolving forest. In such a graph, each node corresponds to a tree in the co-
evolving forest, and two nodes are connected by an edge if there is at least one co-evolution edge between
their corresponding trees. A connected component in the co-evolving forest is a sub-set of trees whose
corresponding nodes in the co-evolutionary graph induce a connected component (see an example in Fig. 1B).
We finish this section with an observation that will be used in the next sections.
Observation 1 (Tuller et al., 2009a). The optimization problem of inferring the ancestral states of a
phylogenetic tree when the optimization criteria is maximum likelihood (Pupko et al., 2000) under i.i.d
models such as Jukes Cantor ( JC) ( Jukes and Cantor, 1969), Neyman (1971), or the model of Yang et al.
(1995) can be formalized as a maximum parsimony problem for non-binary alphabet and with multiple
edge weights (Sankoff, 1975).
(X Y) (Y' Z)
X Y Z
1 1 1 1 1 1 0 0 0 0
x y
x y0 0 1 0 0 1 1 1
weight1000
y z 0 0 1 0 0 1 1 1
0100
0 0
z
0 0
w
1 1
(Z' W')
W
z w 0 0 1 0 0 1 1 1
0001
weight weight
FIG. 2. Reduction from max–2–sat to Ancest–co–evol.
1330 TULLER ET AL.
Observation 1 teaches us that the Ancestral co-evolution problem without co-evolution edges (jEC
(SF)j ¼ 0) can describe a Maximum Likelihood (ML) problem. In this work, indeed the weights of the tree
edges correspond to the probabilities to gain/lose proteins and thus we describe and solve a generalization
of both the ML and the MP problems on trees.
It is important to note that the model that we deal with in this paper is a generalized parsimony and not a
full probabilistic model (like Bayesian network or Markov network). This is still true even when the tree
edges correspond to probabilities of mutation; in this case, when we consider only the tree edge, we solve
the maximum likelihood problem; when we consider also the co-evolutionary edges we can find solutions
that are close to the maximum likelihood solution (depending on the weighting of the co-evolution edges;
see also Subsection 4.2) for the tree edges alone but these are not maximum likelihood solution(s) for a
model that corresponds both to the tree edges and the co-evolutionary edges.
3. HARDNESS ISSUES
In this section, we show that the Ancestral co-evolution problem is NP-hard. We will show it by
reduction from the max–2–sat problem which is known to be NP-hard (Garey and Johnson, 1979) (the
reduction appears in Fig. 2).
Problem 2. Maximum 2-Satisfiability (max–2–sat)
Input: Set U of variables, collection C of clauses over U such that each clause c 2 C has jcj ¼ 2, and a
positive integer K� jCj.Question: Is there a truth assignment for U that simultaneously satisfies at least K of the clauses in C?
Theorem 1. The Ancest–co–evol problem is NP-hard.
Proof. By reduction from max–2–sat. Given an input <U, C, K> to the max–2–sat problem recon-
struct the following input to the Ancest–co–evol problem (Fig. 3): jSFj ¼ jUj, and each tree in SF corre-
sponds to one variable in U; each tree has the same structure and leaf labels as described in Figure 3. Each
co-evolution edge in EC corresponds to one clause in C, and connects the right-most internal nodes in two
trees (Fig. 3). The translation of the four types of possible clauses (true-true, true-false, false-false) to the
weight matrix of its corresponding co-evolution edge appears in Figure 3. Finally, choose
B¼ 2 � jUj þ jCj � jKj.It is easy to see that the optimal parsimony score for each tree in the Ancest–co–evol problem (excluding
the co-evolution edges) is 2; either a solution that labels all the internal nodes with ‘‘0’’ or a solution that
labels all the internal nodes with ‘‘1’’ gives this score.
(X Y) (Y' Z)
X Y Z
1 1 10 0
x y
x y0 0 1 0 0 1 1 1
weight
W2W1W1W1
y z 0 0 1 0 0 1 1 1
W1W2W1W1
0
z
0
w
1
(Z' W')
W
z w 0 0 1 0 0 1 1 1
W1W1W1W2
weight weight
FIG. 3. Reduction from gap–max–2–sat to gap–Ancest–co–evol.
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1331
) Suppose the answer to the max–2–sat problem is YES (i.e., there is a truth assignment for U that
simultaneously satisfies at least K of the clauses in C). In this case, we choose the labeling of all the internal
nodes of each tree to be identical to the assignment of its corresponding variable in U. Thus, the contri-
bution of all the tree edges to the parsimony score (i.e., excluding the co-evolution edges) is 2 � jUj.By the construction of the weight tables (Fig. 3), the contribution to the parsimony score due to
co-evolution edges that correspond to one of the K satisfied clauses is 0, and the contribution to the parsimony
score due to each of the other co-evolution edges is at most 1. Thus, the contribution of all the co-evolution
edges to the parsimony score is at most jCj � jKj, and the answer to the Ancest–co–evol problem is YES.
( Suppose the answer to the Ancest–co–evol problem is YES (i.e., the parsimony score along all the
edges is no more than 2 �jUj þ jCj � jKj). As mentioned earlier, for optimal parsimony score, in each tree all
the internal nodes should be labeled with the same label as that of the internal node that is connected by co-
evolution edges. Thus, the contribution of all the tree edges to the parsimony score is 2 � jUj, and the
contribution of all co-evolution edges to the parsimony score is at most jCj � jKj. Thus, the contribution to
the parsimony score for K of the co-evolution edges is 0. By the reconstruction of the co-evolution edges,
they correspond to K clauses in C that are be satisfied when the assignment to each variable in U is identical
to the labeling of the internal nodes of its corresponding tree. Thus, the answer to the max–2–sat problem is
YES. &
In the rest of this section, we will show that Ancest–co–evol is hard to approximate; i.e., we will show
that the following gap problem is NP-hard:
Problem 3. Gap ancestral co-evolution (gap–Ancest–co–evol[B1, B2])
Input: A co-evolving forest, F¼ (SF, EC(SF)), and two real number, B1 and B2.
Output: If there are labels for the internal nodes of all the trees in the co-evolving forest such that the
sum of the corresponding weights along all the tree edges and the co-evolution edges is less than B1 then
answer ‘‘Yes’’; if there are no labels for the internal nodes of all the trees in the co-evolving forest such that
the sum of the corresponding weights along all the tree edges and the co-evolution edges is less than B2
then answer ‘‘No’’; otherwise answer either ‘‘Yes’’ or ‘‘No.’’
We will show it by reduction from gap–max–2–sat that is known to be NP-hard (Haastad, 2001):
Problem 4. gap–max–2sat[r1*jCj, r2*jCj]Input: Set U of variables, collection C of clauses over U such that each clause c 2 C has jcj ¼ 2, and two
positive real numbers, r1< r2< 1, such that r1*jCj and r2*jCj are integers.
Output: If there is a truth assignment for U that simultaneously satisfies at least r2*C of the clauses in C
then answer ‘‘Yes’’; if no truth assignment for U simultaneously satisfies more than r1*C of the clauses in
C then answer ‘‘No’’; otherwise answer either ‘‘Yes’’ or ‘‘No.’’
By (Hastad, 2001)q2
q1� 22
21.
Theorem 2. The gap–Ancest–co–evol[(jECj–r2*jECj)*W2þ r2*jECj*W1, (jECj � r1*jECj)*W2þr1*jECj*W1] problem is NP-hard.
Proof. Given an input <U, C, K >to the gap–max–2–sat problem an input to the Ancest–co–evol
problem which is similar to the reduction described in theorem 2. The only changes are the topology of the
trees, the weight matrixes, and the fact that there is no penalty on the labels at the ends of the tree edges; see
Figure 3); we assume that W2>W1 and choose B1¼ (jECj � r2*jECj)*W2þ r2*jECj*W1 and
B2¼ (jECj � r1*jECj)*W2þ r1*jECj*W1].
By our reduction, if more than r2*jC j clauses can be satisfied () the Ancest–co–evol has a solution
whose cost <B1. If no more than r1*jCj clauses can be satisfied () the Ancest–co–evol does not have a
solution whose cost<B2. Thus if gap–max–2sat[r1*jCj, r2*jCj] is NP-hard, then gap–Ancest–co–
evol[(jECj � r2*jECj)*W2þ r2*jECj*W1, (jECj � r1*jECj)*W2þ r1* jECj*W1] is NP-hard.
Thus, there is no algorithm with approximation ration 5(1�q1)�W2=W1þ q1
(1�q2)�W2=W1þ q2
. Since W2>W1 the upper
bound on the approximation ratio is gained for very large W2=W1 : (1� q1)(1� q2)
&
Corollary 1. There is a constant z0 such that there is no polynomial time algorithm for Ancest-co-evol
with performance ratio better than z0.
1332 TULLER ET AL.
4. METHODS
4.1. An FPT algorithm and approximation heuristics
As we have shown in the previous section, the Ancestral co-evolution problem is NP-hard. In this
section, we describe an FPT algorithm and an approximation algorithm for the Ancestral co-evolution
problem. The approximation algorithm is described in Figure 4. It has three main steps: (1) The input co-
evolving forest is partitioned into smaller co-evolving forests; (2) optimal labels are assigned to the internal
nodes of each of these co-evolving forests by an FPT algorithm; in total, these labels are an approximation
of the solution for the input co-evolving forest; (3) finally, the solution is further improved greedily.
We will start by describing an FPT algorithm that finds the optimal solution for the Ancestral co-
evolution problem in time complexity that is exponential with the number of trees in SF but polynomial
with the other properties of the input. This algorithm is used in step 2 of the approximation algorithm where
it is implemented on subsets of SF. Similarly to many algorithms for computing the labels of internal tree
nodes (Fitch and Margoliash, 1967; Sankoff, 1975), our algorithm has two phases (in the first phase, it
traverses the co-evolving forest from the leaves to the root; in the second phase, it traverses the co-evolving
forest from the root to the leaves). However, our algorithm is performed jointly for all the trees in each
connected component.
Let Wce;(a, b)¼Wc
(i, j);(a, b) denote the cost of assigning a to node i, and b to node j where (i, j) is a co-
evolution edge. Similarly, if (i, j) is a tree edge we use Wbe;(a, b)¼Wb
(i, j);(a, b) to denote the cost of assigning a
FIG. 4. (A) The general algorithm for inferring ancestral states under co-evolution. (B) The algorithm for partitioning
the co-evolutionary graph. (C) The algorithm for finding the ancestral states of a connected component (output of the
Partite algorithm). (D) The Greedy stage of the algorithm.
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1333
and b to the two nodes of e respectively. Let S�tt(�vv) denote the cost of a sub-forest in the co-evolving forest
whose roots are �tt, when assigning �vv to these roots.
Let tj denote the j-th node in the vector of nodes �tt. Let �kk and �ll denote the corresponding vectors of
children of �tt in the co-evolving forest. In the first phase, all S�tt(�vv) are computed by the following dynamic
programming formula (Fig. 4C):
S�tt(�vv)¼min�uu
XjWb
(tj , lj);(vj , uj)þ S�ll(�uu)
n oþmin�ww
XjWb
(tj , kj);(vj, wj)þ S�kk(�ww)
n oþX
j1, j2:(tj1 , tj2 )2Ec
Wc(tj1 , tj2 );(vj1
, vj2)
In the second phase, the algorithm traverses the sub-forest from the roots to the leaves, and optimal
values are assigned to the internal nodes of the co-evolving forest by the following dynamic programming
formula (Fig. 4C):
For the roots of the co-evolving forest:
�vv� ¼ argmin�vvfS�tt(�vv)gFor a general vector of internal nodes �kk corresponding to the same organism in the co-evolving forest
(after the values �tt� were assigned to its parents, �tt):
�vv � ¼ argmin�vv
XjWb
(tj , kj;(t�j , vj)þ S�kk(�vv)
n o
The running time of this algorithm on a co-evolving forest with m0 trees of size n is O(n � 23�m0 ) since, in
each of the O(n) vectors of internal nodes (nodes corresponding to the same organism) it checks all the 2m0
possible labels of the vector of internal nodes vs. the 2m0· 2m0 possible simultaneous labels to all nodes
corresponding to the vectors of the direct descendant of this vector.
As the running time of the algorithm described above is exponential with the size of co-evolving forest,
the general algorithm (Fig. 4A) has an initial stage (stage 1) where the input graph is partitioned into small
enough connected components.
As the running time of the algorithm described above is exponential with the size of the largest
connected component in the co-evolutionary graph, the general algorithm (Fig. 4A) has an initial stage
where the input graph is partitioned into small enough connected components. The input to the general
algorithm (ACE) includes the maximal size of a connected component after this stage. Let M denote this
parameter.
This step is described in Figure 4B and is performed by an algorithm that recursively implements
k-means (MacQueen, 1967) on the co-evolutionary graph. On the first iteration, the number of clusters is
jVj/M where jVj is the number of vertices in the co-evolutionary graph. If the size of some cluster is larger
than M, the algorithm is executed recursively on this cluster to further partite it to smaller connected
components. The algorithm stops when all parts of the graph (connected components) are smaller than M.
Though the problem of clustering is NP-hard, in practice, and as reported in the next section, the k-means
algorithm is very fast.
The input to this step is a weighted graph whose edges correspond to the edges in the co-evolutionary
graph. The weights of the graph edges can be any measure that represents the strength of the co-evolution
between the corresponding trees (for example, the correlation between the phyletic pattern of the corre-
sponding proteins).
The final step of the Ancestral-Co-Evolver algorithm is a greedy stage (Fig. 4D). In each iteration, the
greedy algorithm searches for an edge and labels its ends in a way that improves the cost of the co-evolving
forest. As demonstrated in the simulations in Section 5.2, this algorithm converge to a local optimum faster
than the running time of the dynamic programming stage with M¼ 7. Note that the greedy algorithm can be
stopped after a certain number of iteration if it does not converges to a local optimum. The greedy stage can
be used as an independent algorithm when running it from various initial points (e.g., one of the initial
points can be the ML solution).
Let kk(SF) denote the labels found by the Ancestral-Co-Evolver algorithm. Let (kk(SF)) denote the par-
simony score induced by these labels. Let MP� (kk(SF)) denote the parsimony score induced by these labels
when not considering the co-evolution edges between the connected components found by the Partite
algorithm; let E� denote this set of co-evolution edges. An upper bound on the approximation ratio of the
general algorithm is given in the following observation:
Observation 2. The approximation ratio of the Ancestral-Co-Evolver algorithm is � MP(kk(SF))
MP� (kk(SF)).
1334 TULLER ET AL.
Proof. The labels found for the output of the Partite algorithm are optimal for that graph (the co-
evolutionary graph without E�), and as this output includes less co-evolution edges than SF, the optimal M
P score is � MP� (kk(SF)). &
We used Observation 2 in order to estimate the approximation ratios of the different algorithms in the
simulations.
One important property of the algorithm is that it enables a trade-off between accuracy and speed. A
larger M (smaller co-evolving sub-forests) increases the running time exponentially but at the same time
increases the accuracy of the solution; M¼ jSFj will give the optimal solution for the Ancestral co-evolution
problem.
Finally, it is important to note that by weighting the co-evolution edges relatively to the tree edges we
can control the relative influence of these two sources of information (co-evolution vs. the evolutionary
tree) on the resulting labels. Thus, for example, it is easy to see that (and a very similar proof can be
outlined for the case where tree edges are rational numbers):
Let kk(SF=EC) denote the set of the optimal labels of SF when not considering the co-evolution edges (i.e.,
EC¼ 0).
Observation 3. If the weight tables of the tree edges are natural numbers and all the entries in the
weight tables of the co-evolution edges < 1jEc(SF )j then the optimal labeling of the co-evolving forest is one of
the optimal labels in kk(SF=EC).
Proof. Suppose a labeling l1 has the optimal cost on the co-evolution forest SF and is not in kk(SF=Ec).
Thus, the contribution of the tree edges to the cost of l1 is greater than those of labelings in kk(SF=Ec). When
considering only the tree edges, the minimal difference between the costs of two solutions that have
different costs is �1. On the other hand, the maximal contribution of all the co-evolution edges to the cost
of any labeling is less than 1. As all weights are positive, l1 cannot be an optimal solution to the complete
co-evolution forest (as any solution in kk(SF=Ec) has a lower cost). &
By Observation 1, the ancestral co-evolution problem without co-evolution edges describes a conven-
tional maximum likelihood problem. Thus, by Observation 3, if we choose small enough weights for
the co-evolution edges our method can be used for choosing one of the optima (or a point very close to
one of the optima) of the maximum likelihood function—the one that is supported by co-evolutionary
relations.
4.2. Weight tables and weighting of the co-evolution edges
A pair of evolutionary trees was connected by a co-evolution edge if the following conditions were
satisfied: (1) we found an evidence for physical or function interaction between the two corresponding
proteins (based on String database (2) The co-occurrences distribution of the two proteins in a large group
of organisms (we used the data from Tuller et al., 2009a) was far from being uniform (see an example in
Section 5.3). The values in the co-evolutionary weight tables were based on the co-occurrences distribution
in the organisms from Tuller et al. (2009a) (i.e., each table included the four possible probabilities that the
corresponding labels of the pair of proteins will appear in an organism).
We tested several values for the weights of the co-evolution edges compared to the tree edges. At one
extreme, the entries of the tree edges weight tables are multiplied by a very large constant. In this case,
the tree edge weight tables are dominant compared to the weights of the co-evolution edges (the solution is
one of the ML/MP solutions). At the second extreme, the fifth weighting, the co-evolution edge weights are
dominant compared to the tree edge weights. In this case, the entries of the tree edges weight tables are
multiplied by a very large constant.
Let MPb(SF, W) denote the parsimony score when solving the ancestral co-evolution problem with
weighting W and when considering only tree edges. Let MPc(SF, W) denote the parsimony score when
solving the ancestral co-evolution problem with weighting W and when considering only co-evolution
edges. In this work, we used the weighting, W*, that optimizes the sum of the two sources of information
(co-evolution, and the evolutionary trees); i.e., we used W� ¼ argminW
�MPc(SF , W)
minW (MPc(SF , W))þ MPb(SF , W)
minW (MPb(SF , W))
�.
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1335
5. EXPERIMENTAL RESULTS
5.1. Simulated evolution
To analyze the performances of the algorithms described in the previous section we generate a proba-
bilistic process that describes the evolution of a co-evolving forest. In the simulation, each character
evolves along the branches of the evolutionary trees, but also has correlations with the other characters that
interact with it.
The simulation was performed as in Tuller et al. (2009a): The topology of the trees in the co-evolving
forest, their edges’ lengths (probability of mutation according to JC model), and the co-evolution edges
between pairs of nodes at the root of the trees, were all chosen randomly. We also assigned an additional
parameter, pc, to each pair of nodes that are co-evolutionary legal. This parameter denotes a small
probability that a co-evolution edge between the node pair will appear/disappear. For simplicity we assume
a binary alphabet.
To simulate the co-evolutionary process we generated a probabilistic process where each character
evolves along the branches of the evolutionary trees (stage 1), but also has correlations with the other
characters that interact with it (stage 2). The process has two main stages that are performed sequentially:
Stage (1) (Fig. 5): The label of a node is generated after the label of its ancestor was generated (according to
the corresponding tree branch length). The initial node is the root. Stage 2 (Fig. 5): After the labels of the
nodes corresponding to all the sites of a certain organism in the co-evolving forest are generated, the label
of each node is switched according to those of its neighbors (i.e., nodes in other trees that are connected to
it with co-evolution edges). The switching is performed by randomly traversing all the nodes in the network
and setting the label of each node to a value (‘‘0’’ or ‘‘1’’) that appears in the majority of its neighbors (the
value of a node cannot be changed after it was set).
In each experiment, the input to the algorithms included the following: (1) The real tree topology. (2)
A noisy version of the edge lengths; we translate the input to a weight table in the following way: let pe
denote the mutation probability along the tree edge e; the weight entry of the corresponding edge
is� log( pe) if the labels at the ends of the edge are equal, and� log(1� pe) if the labels at the ends of the
edge are not equal. (3) As co-evolution edges we took the set of the co-evolution edges that was generated
between the nodes corresponding to one of the organisms at the leaves (Fig. 5C). (4) The weight entries
of the co-evolution edges were computed in each experiment by the phyletic patterns of the corre-
FIG. 5. Simulation of co-evolution. (A) The two major steps: (1) Red (first stage): evolution according to the
evolutionary trees. (2) Blue (second stage): selection according to the interacting sites (co-evolution). (B) Stage (2)
involves a random walk along all the nodes corresponding to a certain organism in the co-evolving forest; the value of a
node is set to be similar to the value in the majority of its neighbors. (C) At the beginning of the simulation, random co-
evolution edges, between pairs of roots in the co-evolving forest are added. There is a small probability that some of
these initial co-evolution edges will be lost or that new ones will emerge. (D) Average probability to get the same label
in a pair of nodes in this simulation as a function of the distance in the co-evolutionary graph and in the phylogenetic
tree. Nodes that are closer in the co-evolutionary graph or in the phylogenetic tree tend to be more similar.
1336 TULLER ET AL.
sponding pairs of nodes at the leaves. Let pa,b denote the empirical probability that a pair of labels (a, b)
for the nodes connecting a co-evolution edges appears at the leaves. The corresponding weight for this
entry is� log( pa,b).
5.2. Simulation results
We compared the performances of the following algorithms: (1) The Partitioning algorithm (Fig. 4B)
with the Dynamic Programming algorithm (Fig. 4C). (2) The greedy algorithm (Fig. 4D.) with a few initial
points (one of them is the the ML solution). (3) The ACE algorithm (all the stages in Fig. 4). (4) The ML
and MP algorithms that do not consider the co-evolution edges. Let DP X denotes a Dynamic Programming
algorithm with M¼X ; let ACEX denotes an ACE algorithm with M¼X.
A summary of the simulation results appears in Figure 6; sub-figures A–C depict the running times (log
scale) and sub-figures D–F describe the approximation ratios as functions of the size of the evolutionary
FIG. 6. Running time and accuracy of the partitioning algorithm with the Dynamic Programming algorithm, the
greedy algorithm, the ACE, and the ML/MP algorithms. (A–C) The log running time (in ms) of the algorithms as
function of the size of the evolutionary trees (A; 200 trees and 2.5 co-evolution edges per node in the co-evolutionary
graph), number of co-evolutionary trees (B; 2.5 co-evolution edges per node in the co-evolutionary graph, each
evolutionary tree with 70 leaves), and the number of co-evolution edges per node in the co-evolutionary graph (C; 200
evolutionary trees, each with 70 leaves). (D–F) The upper bound on the approximation ratio (Observation 2) of the
solution found by each of the algorithms as function of the number of leaves in each evolutionary tree (D), the number
of evolutionary trees (E), and the number of co-evolution edges per node in the co-evolutionary graph (F).
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1337
trees, the number of evolutionary trees, the number of co-evolution edges per node in the co-evolutionary
graph. All the running finished in less than a four minutes. As can be seen, the running time increases
exponentially with M (see ACE7, DP7 vs. ACE2, DP2 sub-figures A–C) while the running time of the
greedy algorithm alone is larger than DP2 but exponentially smaller than DP7. As can be seen, in the case
of running time, the most influential parameter is the number of evolutions trees in the input.
In the case of the approximation ratio (the upper bound from Observation 2), the most influential
parameter is number of co-evolution edges per node in the co-evolutionary graph. As can be seen, ACE7
performs better than all the other algorithms and always has approximation ratio of <1.3. Interestingly, the
greedy algorithm is only a few percentages worse. These results support using the greedy algorithm if
running time matters. The fact that the upper bound of the ACE7< 1.3 demonstrates that our approach can
find solutions that are very close to the optimal ones. As we used here an upper bound on the approximation
ratio the actual ratio can be significantly lower.
In the simulation, as in the case of biological data (see the next section), the ML solutions (when ignoring
co-evolution) are relatively similar to the solutions found by our approach. Thus, it is not surprising that
approximation ratio of ML is bound to be <1.7. On the other hand, the margin between the approximation
ratio of the ML and that of the ACE is significant: up to 30%. This result demonstrates the essentiality of
our approach.
5.3. A biological example: reconstruction of the ancestral genome content the Fungi
Using the method outlined above we set to reconstruct the ancestral genome content of 17 Fungi whose
evolutionary tree appears in Figure 7. The input included 33,931 families of Fungi orthologs (downloaded
from Wapinski et al., 2007) and a total of 20,317 co-evolution edges.
We represented each of the 17 genomes by a binary string of length 33931 where ‘1’ in the x position of a
string means that there is a gene/protein from the x group of orthologs in this genome, and ‘0’ means that
there is no gene/protein from the x group of orthologs in this genome. We used Neyman’s two state model
(Neyman, 1971), a version of Jukes Cantor ( JC) model ( Jukes and Cantor, 1969) for inferring the edge
S. pombe
A. nidulans
M. grisea
F. graminearum
N. crassa
Y. lipolytica
C. albicans
C. hamasenii
K. waltii
A. gossypii
K. lactis
S. castellii
C. glabrata
C. bayanus
S. mikatae
S. paradoxus
S. Cerevisiae
15
14
2
1 0
13 3
6
7
8
9
4 Hemiascomycota
Euascomycota
Archeascomycota
125
10
11
A
100
The no. of proteins inferred by ACE vs.the no. of proteins inferred by ML/MP
20
40
60
80
2 4 6 8 10 12 14
Found only by ACEFound only by ML
ML
Dif
f. in
th
e n
o. o
fin
ferr
ed p
rote
ins
Dif
f. in
th
e n
o. o
fin
ferr
ed p
rote
ins
Index of internal node
0 2 4 6 8 10 12 140
20
40
60
80
100
120
140
Found only by ACEFound only by MP
MP
B
C
0
FIG. 7. (A) The evolutionary tree of the Fungi that was analyzed by the ACE. (B) Summary of the results for the MP
case; the number of proteins inferred by ACE and not by MP and vice versa. (C) Summary of the results for the ML
case; the number of proteins inferred by ACE and not by ML and vice versa.
1338 TULLER ET AL.
lengths of the tree by maximum likelihood. This was done by PAML (Yang, 1997). These edge lengths
correspond to the probabilities that a protein will appear/vanish along the corresponding lineage.
As co-evolution edges we use various physical and functional interactions that were downloaded from
String et al. (2009) (http://string.embl.de/) (Tuller et al., 2009b) reported the relation between co-evolution
and similar functionality). We filtered co-evolutionary edges for which the ratio between the highest and
lowest probability in the co-occurrence distribution table was less than 4.25. The weights in the tables were
computed according to the co-occurrence probabilities of the corresponding pairs of orthologs. We used the
weighting whose corresponding solution optimizes both the score of the co-evolution edges and the score of
the tree edges (see more details in Subsection 4.2). The annotation of ancestral proteins was based on the
GO annotations of S. cerevisiae.
We compared our results of the ACE to those obtained by using only tree edges (by using MP and ML).
The total running time of the ACE algorithm on this large biological dataset was about 1.5 hours on a
conventional PC. Figure 7 summarizes the results of the analysis by ACE. As can be seen, ACE removed/
added hundreds of proteins to ML/MP labels of the internal nodes of the evolutionary trees. A major
fraction of the discrepancies between the results of the ACE algorithm and those obtained with the
conventional methods (ML and MP) appears in the Euascomycota subtree (internal nodes 0–2 in the Fungi
tree; Fig. 7A). In general, ACE mainly added proteins to these nodes, implying that ML/MP underestimated
the size of these ancestral genomes. Additionally, both in the case of MP and ML, the ACE added/removed
many proteins from internal nodes 14 and 15. It is important to note that the likelihood score of the ACE
solution was only 0.25% lower than the ML score, demonstrating that this solution was very close to the
ML point. Similarly, when the ACE was implemented with MP the parsimony score of its solution was only
2.2% higher than that of the MP solution. These solutions, however, are supported by the co-evolutionary
information and thus are more biologically plausible.
We further analyzed the proteins added by the ACE to the ML solution for the ancestors of the
Euascomycota (internal nodes 0–2): the nodes with the largest number of discrepancies between ML and
ACE. The groups of proteins added to each of these nodes were very similar (around 95% similarity); thus
we report only the results for node 2, the ancestor of the Euascomycota.
ACE added 89 proteins to the ML solution of internal node 2. Various pieces of evidence support the
biological plausibility of the addition of these proteins by ACE: First, the group of proteins added to this
node was enriched with proteins that take part in basic and essential metabolic processes. Specifically, it
was enriched with the cellular process: protein amino acid phosphorylation (p-value¼ 0.00054), amine
transport (p-value¼ 0.00153), and amino acid transport (p-value¼ 0.00518); all p-values passed the False
Discovery Rate (FDR) control for multiple hypothesis testing (Benjamini and Hochberg, 1995). Second, all
these proteins have orthologs in most of the analyzed Fungi (on average in 76% of the Fungi); this fact also
supports the essentiality of these proteins. Third, in S. cerevisiae, many of these proteins are part of the
same complex with proteins inferred by the standard ML (note that co-evolutionary relations used by ACE
do not necessarily imply association in the same complex).
The following are two typical examples that further demonstrate the three points mentioned above:
Example 1. Three of the proteins added by the ACE are orthologs of S. cerevisiae TPK1, TPK2, and
TPK3. The presence of at least one of these genes is required for normal growth in S. cerevisiae (Toda et al.,
1987). These genes are part of the cAMP-dependent protein kinase complex that also includes another
protein (BCY1), which was inferred by ML.
Example 2. Orthologs of the S. cerevisiae FAT1 gene appear in 76% of the analyzed Fungi. In
S. cerevisiae, this protein forms a complex with FAA1 or FAA4 that imports and activates exogenous fatty
acids (Zou et al., 2002). Both FAA1 and FAA4 were inferred by ML.
5.4. Reconstructs missing values at the leaves of the evolutionary tree by the ACE
At the next first stage, to further demonstrate the advantages of ACE over conventional ML/MP and to
study how co-evolutionary relations improve error rate we performed the following procedure: (1) We
randomly flipped 1.5%–4.5% of the values (500–1500 sites) with co-evolutionary relations in 6%–18% of
the genomes (1–3 genomes), from absence to presence or vice versa. (2) We reconstructed the ancestral
states of the co-evolutionary forest based on the altered genomic contents. (3) We then ‘‘fixed’’ the values
at the leaves that were flipped by choosing labels that optimize the score of the co-evolutionary network
given the states inferred in (2). (4) Steps 1–3 were repeated 7 times and the resulting error-rates were
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1339
averaged (to decrease the bias; small changes in the number of repeats do not change the conclusion of this
analysis).
We considered three different modes of samplings of the sites to be flipped: (1) Coev only—sample only
values that have co-evolutionary relations. (2) Uniform—sample uniformly from all values. (3) No Coev—
sample only values that do not have co-evolutionary relations.
The mean error-rate of the inferred values at the leaves for the above procedure and for different MP/CE
or ML/CE weightings, different sampling modes, MP and ML, as well as for different percent of flipping is
provided in Table 1.
As can be seen, in the case of Coev only sampling, using the co-evolutionary information reduces the
error rate by more than 50% where usually most of the improvement is achieved in the case of the first or
the second weighting (whose MP/ML score is also relatively high). In the case of the Uniform sampling
most of the improvement is also achieved in the case of the first or the second weighting but the im-
provement is more modest. As can be seen, in the case of the No Coev sampling, the error rate was much
higher (more than 500% higher) than in cases where we sampled values with co-evolutionary relations.
Finally, usually the weighting that optimized the error-rate included the two sources of information (co-
evolution, and the evolutionary trees), demonstrating that they are both important.
This analysis further supports the use of co-evolutionary information on top of conventional ML/MP
approaches.
6. CONCLUSION
In this study, we formally described a new computational approach for reconstructing ancestral genomic
sequences using information about co-evolution. Our model captures co-evolutionary dependencies be-
tween different proteins and uses this information to disambiguate the labels of the reconstructed ancestral
genomes. We showed that this computational problem is NP-hard, and described algorithms for solving it.
We demonstrated the performances of our approach by analyzing simulated input and by analyzing the
ancestral genome content of the Fungi, showing that our approach can be used in practice and that it
outperforms the convention ML/MP approaches.
In the future we intend to generalize this work in several ways. First, currently, our approach is presented
in the setup of ancestral genome reconstruction, due to the importance of this problem and because co-
Table 1. Reconstructs Missing Values at the Leaves of the Evolutionary Tree by the ACE
Mode
Sample
mode
Sampling size
organisms
Sampling
size sites i.i.d.
Coev
norm 1
Coev
norm 2
Coev
norm 3
Coev
norm 4
Coev
norm 5
Coev
only
ML Coev only 1 500 0.043 0.043 0.014# 0.0134 0.00914 0.00714* 0.00714
ML Coev only 2 1000 0.068 0.068 0.024# 0.0212 0.0202 0.0159 0.0154*
ML Coev only 3 1500 0.107 0.107 0.0438# 0.04 0.039 0.035* 0.036
ML Uniform 1 500 0.116 0.116 0.113# 0.113 0.113 0.11285* 0.932
ML Uniform 2 1000 0.142 0.142 0.138# 0.138 0.1378 0.137* 0.934
ML Uniform 3 1500 0.186 0.186 0.179# 0.179 0.179 0.1787* 0.933
ML No Coev 1 500 0.11*,# 0.11 0.11 0.11 0.11 0.11 1
ML No Coev 2 1000 0.147*,# 0.147 0.147 0.147 0.147 0.147 1
ML No Coev 3 1500 0.192*,# 0.192 0.192 0.192 0.192 0.192 1
MP Coev only 1 500 0.099 0.0365# 0.0268 0.0186 0.0134 0.00485 0.00428*
MP Coev only 2 1000 0.183 0.068# 0.0433 0.0369 0.028 0.0146 0.0146
MP Coev only 3 1500 0.288 0.1196# 0.073 0.0594 0.0472 0.0341* 0.0348
MP Uniform 1 500 0.146 0.1437# 0.1437 0.14285 0.141* 0.141 0.934
MP Uniform 2 1000 0.176 0.168 0.166 0.165 0.165 0.164 0.937
MP Uniform 3 1500 0.237 0.2255# 0.2219 0.22 0.2188 0.2176* 0.934
MP No Coev 1 500 0.14*,# 0.14 0.14 0.14 0.14 0.14 1
MP No Coev 2 1000 0.18*,# 0.18 0.18 0.18 0.18 0.18 1
MP No Coev 3 1500 0.23*,# 0.23 0.23 0.23 0.23 0.23 1
#, the maximal decrease in the error-rat; *, the minimal error rate.
1340 TULLER ET AL.
evolutionary information can be readily obtained on the gene/protein level. However, it is important to note
that the potential scope of our approach goes beyond the ancestral genome reconstruction problem, to tackle
the more general problem of ancestral sequence reconstruction: that is, the reconstruction of different sites or
domains in proteins, and even (provided that sufficient information is available) the reconstruction of single
sites in DNA or RNA sequences (Yeang et al., 2007; Yeang and Haussler, 2007; Lockless and Ranganathan,
1999; Pedersen et al., 2006; Knudsen and Hein, 1999; Rzhetsky, 1995). In this setup, the success of such
future applications depends on the existence of reliable co-evolutionary information on the individual site
level. For example, information about the secondary structure of RNA sequences (Cannone et al., 2002) and
proteins (Kabsch and Sander, 1983) can be used for determining what pairs of sites co-evolve.
Second, it is also clear that our approach can be generalized in the future to more complex reconstruction
models for example, using non-binary alphabets (Tuller et al., 2009a), dependency between close sites, and
various versions of maximum likelihood. Third, we intend to design algorithms that may compete with
those described in this work. Specifically, we intend to check algorithms that are based on the belief
propagation (Kschischang et al., 2001) approach. Fourth, we believe that a generalization of the approach
described in this work can be used for joint inference of ancestral genomes and protein interactions or for
joint inference of ancestral genomes and metabolic networks of ancestral and extant organisms.
Fifth, as we mentioned in Section 2, the model that we deal with in this article is a generalized parsimony
and not a full probabilistic model (like Bayesian network or Markov network). Another open computational
direction is to design a full probabilistic model for the problem of ancestral reconstruction that will take
into account co-evolutionary relations.
Finally, in this study we assume that different proteins have an identical phylogenetic tree. In general,
this assumption is not true; due to horizontal gene transfer (HGT) (Doolittle and Bapteste, 2007) and gene
duplication events different orthologs may have different phylogenetic trees (Wapinski et al., 2007).
However, at the level of entire genomes the ‘‘signal’’ of one (or a small number of ) phylogenetic tree (the
‘‘tree of life’’) usually emerge (Ulitsky et al., 2006; Ge et al., 2005; Puigbo et al., 2009). In this study, we
indeed deal with the organismal level: reconstruction of the gene content of ancestral genomes. We believe
that co-evolutionary constraints play an important role also in the case of horizontal gene transfer (similarly
to the cases of vertical inheritance). Thus, our approach should be even more useful when the analyzed
organisms underwent many horizontal gene transfer events and/or gene duplications and there is no
information about the different tree topologies of the different protein families.
Suppose the different protein families have different tree topologies and these different topologies are
given. If we know how to embed the protein trees in the organisms tree we actually also know the solution
of the ancestral reconstruction problem (the number of proteins from each family in each ancestral or-
ganism; see Fig. 8). If the embedding is not known, we believe that usage of co-evolutionary information
can be useful also for solving the embedding problem (Page and Charleston, 1997)). Co-evolutionary
Organism tree
Gene tree -
protein family
Genome A Genome B Genome C
Ancestral
Genome D
Ancestral
Genome E
a b c d
Duplication
Loss
HGT
a b c d
Reconciliation
FIG. 8. The problem of embedding a gene tree in a species tree (or the tree reconciliation problem). In this problem
we seek an explanation to the differences between the two trees (the gene tree and the species tree). In this figure, a gene
tree with four leaves (a, b, c, d) is embedded in a species tree with three leaves (A, B, C), and two internal nodes (D, E).
As can be seen, the solution to the embedding problem includes the solution to the ancestral genome problem (the
number of proteins from each gene family in each ancestral organism); however, it includes additional information (full
explanation to the differences between the gene tree and the species tree).
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1341
information can add constraints that are related to the number of proteins from each family in each ancestral
organism; thus, it can ‘‘guide’’ the algorithms that search the full explanation for the evolution of a gene
family within the species tree.
ACKNOWLEDGMENTS
T.T. is a Koshland Scholar at Weizmann Institute. M.K. and E.R. were supported by grants from the
Israel Science Foundation.
DISCLOSURE STATEMENT
No competing financial interests exist.
REFERENCES
Barry, D., and Hartigan, J. 1987. Statistical analysis of humanoid molecular evolution. Stat. Sci. 2:191–210.
Benjamini, Y., and Hochberg, Y. 1995. Controlling the false discovery rate: a practical and powerful approach to
multiple testing. J. R. Statist. Soc. 57:289–300.
Blanchette, M., Green, E.D., Miller, W., et al. 2004. Reconstructing large regions of an ancestral mammalian genome in
silico. Genome Res. 14:2412–2423.
Cai, W., Pei, J., and Grishin, N.V. 2004. Reconstruction of ancestral protein sequences and its applications. BMC Evol.
Biol. 4:e33.
Cannone, J.J., Subramanian, S., Schnare, M.N., et al. 2002. The comparative RNA web (CRW) site: an online database
of comparative sequence and structure information for ribosomal, intron, and other rnas. BioMed Central Bioin-
formatics 3, 15.
Chor, B., Hendy, M.D., Holland, B.R., et al. 2000. Multiple maxima of likelihood in phylogenetic trees: an analytic
approach. Mol. Biol. Evol. 17:1529–1541.
Cohen, O., Rubinstein, N.D., Stern, A., et al. 2008. A likelihood framework to analyse phyletic patterns. Phil. Trans. R.
Soc. Lond. B. Biol. Sci. 363:3903–3911.
Csuros, M., and Miklos, I. 2009. Streamlining and large ancestral genomes in archaea inferred with a phylogenetic
birth-and-death model. Mol. Biol. Evol. 26:2087–2095.
Doolittle, W.F., and Bapteste, E. 2007. Pattern pluralism and the tree of life hypothesis. Proc. Natl. Acad. Sci. U.S.A.
104:2043–2049.
Elias, I., and Tuller, T. 2007. Reconstruction of ancestral genomic sequences using likelihood. J. Comput. Biol.
14:216–237.
Barker, D., et al. 2007. Constrained models of evolution lead to improved prediction of functional linkage from
correlated gain and loss of genes. Bioinformatics 23:14–20.
Pazos, F., et al. 1997. Correlated mutations contain information about protein-protein interaction. J. Mol. Biol.
271:511–523.
Wu, J., et al. 2003. Identification of functional links between genes using phylogenetic profiles. Bioinformatics
19:1524–1530.
Marino-Ramirez, L., et al. 2006a. Co-evolutionary rates of functionally related yeast genes. Evol. Bioinform. 2295–
2300.
Jensen, L.J., et al. 2009. String 8–a global view on proteins and their functional interactions in 630 organisms. Nucleic
Acids Res. 37:D412–D416.
Chena, Y., et al. 2006b. The coordinated evolution of yeast proteins is constrained by functional modularity. Trends
Genet. 22:416–419.
Felder, Y., and Tuller, T. 2008. Discovering local patterns of co-evolution. Proc. RECOMB-CG 55–71.
Felsenstein, J. 1993. Phylip (phylogeny inference package) version 3.5c. Technical report. Department of Genetics,
University of Washington, Seattle.
Fitch, W. 1971. Toward defining the course of evolution: minimum change for a specified tree topology. Syst. Z.
20:406–416.
Fitch, W.M., and Margoliash, E. 1967. Construction of phylogenetic trees. Science 155:279–284.
Garey, M.R., and Johnson, D.S. 1979. Computer and Intractability. Bell Telephone Laboratories, New York.
Gaucher, E.A., Thomson, J.M., Burgan, M.F., et al. 2003. Inferring the palaeoenvironment of ancient bacteria on the
basis of resurrected proteins. Nature 425:285–288.
1342 TULLER ET AL.
Ge, F., Wang, L., and Kim, J. 2005. The cobweb of life revealed by genome-scale estimates of horizontal gene transfer.
PLoS Biol. 3.
Hacia, J.G., Fan, J.B., Ryder, O., et al. 1999. Determination of ancestral alleles for human single-nucleotide poly-
morphisms using high-density oligonucleotide arrays. Nat. Genet. 22:164–167.
Hastad., J. 2001. Some optimal inapproximability results. J. ACM 48:798–859.
Hillis, D.M., Huelsenbeck, J.P., and Cunningham, C.W. 1994. Application and accuracy of molecular phylogenies.
Science 264:671–677.
Hudek, A.K., and Brown, D.G. 2005. Ancestral sequence alignment under optimal conditions. BMC Bioinform. 6:273.
Jermann, T.M., Opitz, J.G., Stackhouse, J., et al. 1995. Reconstructing the evolutionary history of the artiodactyl
ribonuclease superfamily. Nature 374:57–59.
Jin, G., Nakhleh, L., Snir, S., et al. 2006. Maximum likelihood of phylogenetic networks. Bioinformatics 22:2604–2611.
Juan, D., Pazos, F., and Valencia, A. 2008. High-confidence prediction of global interactomes based on genome-wide
coevolutionary networks. Proc. Natl. Acad. Sci. U.S.A. 105:934–939.
Jukes, T.H., and Cantor, C.R. 1969. Evolution of protein molecules. In Munro, H.N., ed. Mammalian Protein Meta-
bolism. Academic Press, New York.
Kabsch, W., and Sander, C. 1983. Dictionary of protein secondary structure: pattern recognition of hydrogen-bonded
and geometrical features. Biopolymers 22:2577–2637.
Knudsen, B., and Hein, J. 1999. RNA secondary structure prediction using stochastic context-free grammars and
evolutionary history. Bioinformatics 15:446–454.
Koshi, M., and Goldstein, R. 1996. Probabilistic reconstruction of ancestral protein seuences. JME 42:313–320.
Krishnan, N.M., Seligmann, H., Stewart, C. et al. 2004. Ancestral sequence reconstruction in primate mitochondrial
dna: Compositional bias and effect on functional inference. MBE 21:1871–1883.
Kschischang, F.R., Frey, B.J., and Loeliger, H. 2001. Factor graphs and the sum-product algorithm. IEEE Trans.
Inform. Theory 47:498–519.
Li, G., Steel, M., and Zhang, L. 2008. More taxa are not necessarily better for the reconstruction of ancestral character
states. System. Biol. 57:647–653.
Lockless, S.W., and Ranganathan, R. 1999. Evolutionarily conserved pathways of energetic connectivity in protein
families. Science 286:295–299.
Ma, J., Zhang, L., Suh, B.B., et al. 2006. Reconstructing contiguous regions of an ancestral genome. Genome Res.
16:1557–1565.
MacQueen, J.B. 1967. Some methods for classification and analysis of multivariate observations. Proc. 5th Berkeley
Symp. Math. Statist. Probabil. 1:281–297.
Neyman, J. 1971. Molecular studies of evolution: A source of novel statistical problems, 127. In Gupta, S., and Jackel,
Y., eds. Statistical Decision Theory and Related Topics. Academic Press, New York.
Ouzounis, C.A., Kunin, V., Darzentas, N., et al. 2006. A minimal estimate for the gene content of the last universal
common ancestor–exobiology from a terrestrial perspective. Res. Microbiol. 157:57–68.
Page, R.D., and Charleston, M.A. 1997. From gene to organismal phylogeny: reconciled trees and the gene tree/species
tree problem. Mol. Phylogenet. Evol. 7:231–240.
Pagel, M. 1999. The maximum likelihood approach to reconstructing ancestral character states of discerete characters
on phylogenies. System. Biol., 48:612–622.
Pedersen, J.S., Bejerano, G., Siepel, A., et al. 2006. Identification and classification of conserved rna secondary
structures in the human genome. PLoS. Comput. Biol. 2:e33.
Puigbo’, P., Wolf, Y.I., and Koonin, E.V. 2009. Search for a ‘‘tree of life’’ in the thicket of the phylogenetic forest.
J. Biol. 8:59.
Pupko, T., Peer, I., Shamir, R., et al. 2000. A fast algorithm for joint reconstruction of ancestral amino acid sequences.
Mol. Biol. Evol. 17:890–896.
Rascola, V.L., Pontarottia, P., and Levasseura, A. 2007. Ancestral animal genomes reconstruction. Curr. Opin. Im-
munol. 19:542–546.
Rzhetsky, A. 1995. Estimating substitution rates in ribosomal RNA genes. Genetics 141:771–783.
Sankoff, D. 1975. Minimal mutation trees of sequences. SIAM J. Appl. Math. 28:35–42.
Sato, T., Yamanishi, Y., Kanehisa, M., et al. (2005). The inference of proteinprotein interactions by co-evolutionary
analysis is improved by excluding the information about the phylogenetic relationships. Bioinformatics 21:3482–
3489.
Tauberberger, J.K., Reid, A.H., Lourens, R.M., et al. 2005. Characterization of the 1918 influenza virus polymerase
genes. Nature 437:889–893.
Thornton, J.W., Need, E., and Crews, D. 2003. Resurrecting the ancestral steroid receptor: ancient origin of estrogen
signaling. Science 301:1714–1717.
Toda, T., Cameron, S., Sass, P., et al. 1987. Three different genes in S. cerevisiae encode the catalytic subunits of the
cAMP-dependent protein kinase. Cell 50:277–287.
RECONSTRUCTING ANCESTRAL GENOMIC SEQUENCES BY CO-EVOLUTION 1343
Tuller, T., Birin, H., Gophna, U., et al. 2009a. Reconstructing ancestral gene content by co-evolution. Genome Res. Nov
30. [Epub ahead of print].
Tuller, T., Kupiec, M., and Ruppin, E. 2009b. Co-evolutionary networks of genes and cellular processes across fungal
species. Genome Biol. 10.
Ulitsky, I., Burstein, D., Tuller, T., et al. 2006. The average common substring approach to phylogenomic recon-
struction. J. Comput. Biol. 13:336–350.
Wapinski, I., Pfeffer, A., Friedman, N., et al. 2007. Natural history and evolutionary principles of gene duplication in
fungi. Nature 449:54–61.
Yang, Z. 1997. PAML: a program package for phylogenetic analysis by maximum likelihood. Comput. Appl. BioSci.
13:555–556.
Yang, Z., Kumar, S., and Nei, M. 1995. A new method of inference of ancestral nucleotide—and amino acid sequences.
Genetics 141:1641–1650.
Yeang, C.H., and Haussler, D. 2007. Detecting coevolution in and among protein domains. PLoS Comput. Biol. 3:e211.
Yeang, C.H., Darot, J.F., Noller, H.F., et al. 2007. Detecting the coevolution of biosequences—an example of RNA
interaction prediction. Mol. Biol. Evol. 24:2119–2131.
Yu Gorbunov, K., Lyubetskaya, E.V., Asarin, E.A., and Lyubetsky, V.A. 2009. Modeling evolution of the bacterial
regulatory signals involving secondary structure. Mol. Biol. 43.
Zhang, J., and Rosenberg, H.F. 2002. Complementary advantageous substitutions in the evolution of an antiviral rnase
of higher primates. Proc. Natl. Acad. Sci. USA. 99:5486–5491.
Zou, Z., DiRusso, C.C., Ctrnacta, V., et al. 2002. Fatty acid transport in Saccharomyces cerevisiae: directed muta-
genesis of fat1 distinguishes the biochemical activities associated with fat1p. J. Biol. Chem. 277:31062–31071.
Address correspondence to:
Dr. Tamir Tuller
Faculty of Mathematics and Computer Science
Wei-zmann Institute of Science
Rehovot, Israel
E-mail: tamir.tuller@weizmann.ac.il
1344 TULLER ET AL.
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